Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-01T06:30:32.831Z Has data issue: false hasContentIssue false
  • English
  • Français

On Characterizations of Linear Groups III

Published online by Cambridge University Press:  22 January 2016

Michio Suzuki
Michio Suzuki*
Affiliation:
University of Illinois, University of Chicago
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In his address at the International Congress of Mathematicians at Amsterdam [1] Professor R. Brauer proposed a problem of characterizing various groups of even order by the properties of the involutions contained in these groups and he gave characterizations of the general projective linear groups of low dimensions along these lines. The detail of the one-dimensional case has been published in [5], but the two-dimensional case has not appeared yet in detail. His work was followed by Suzuki [7], Feit [6] and Walter [11]. The present paper is a continuation of [73 and discusses a characterization of the two-dimensional projective unitary group over a finite field of characteristic 2. The precise conditions which characterize the group in question will be stated in the first section. The method employed here is similar to the one used in [7]. An application of group characters provides a formula for the order. However a difficulty comes in when one attempts to identify the group. In order to overcome this difficulty we will use a method primarily designed to study a class of doubly transitive permutation groups (cf. [9]). We need also a group theoretical characterization of a class of doubly transitive groups called (ZT)-groups. This is a generalization of a result in [8], and may be of in dependent interest.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 21 , December 1962 , pp. 159 - 183
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1962

References

[1] Brauer, R., On the structure of groups of finite order, Proceedings of the International Congress of Mathematicians, vol. 1 (1954), pp. 1–9.Google Scholar
[2] Brauer, R. and Fowler, K. A., On groups of even order, Ann. of Math., vol. 62 (1955), pp. 565–583.Google Scholar
[3] Brauer, R. and Nesbitt, C., On the modular characters of groups, Ann. of Math., vol. 42 (1941), pp. 556–590.Google Scholar
[4] Brauer, R. and Suzuki, M., On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. of the National Acad. Sci. USA, vol. 45 (1959), pp. 1757-1759.Google Scholar
[5] Brauer, R., Suzuki, M. and Wall, G. E., A characterization of the one-dimensional unimodular projective groups over finite fields, Illinois Jour, of Math., vol. 2 (1958), pp. 718–745.Google Scholar
[6] Feit, W., A characterization of the simple groups SL(2, 2α), Amer. Jour, of Math., vol. 82 (1960), pp. 281–300.Google Scholar
[7] Suzuki, M., On characterizations of linear groups, I and II, Trans. Amer. Math. Soc., vol. 92 (1959), pp. 191–219.Google Scholar
[8] Suzuki, M., Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc. vol. 99 (1961), pp. 425–470.Google Scholar
[9] Suzuki, M., On a class of doubly transitive groups, Ann. of Math. vol. 75 (1962), pp. 105–145.Google Scholar
[10] Thompson, J. G., Finite groups with fixed-point-free automorphisms of prime order, Proc. of the National Acad. Sci. USA, vol. 45 (1959), pp. 578–581.Google Scholar
[11] Walter, J. H., On the characterization of linear and projective linear groups, I and II, Trans. Amer. Math. Soc., vol. 100 (1961), pp. 481–529 and vol. 101 (1961), pp. 107–123,Google Scholar